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\begin{document}

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\section*{universe in a topos}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{topos_theory}{}\paragraph*{{Topos Theory}}\label{topos_theory}

[[!include topos theory - contents]]

\hypertarget{universes}{}\paragraph*{{Universes}}\label{universes}

[[!include universe - contents]]

\hypertarget{foundations}{}\paragraph*{{Foundations}}\label{foundations}

[[!include foundations - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{example_universes_in_}{Example: universes in $SET$}\dotfill \pageref*{example_universes_in_} \linebreak
\noindent\hyperlink{axioms_of_universes}{Axioms of universes}\dotfill \pageref*{axioms_of_universes} \linebreak
\noindent\hyperlink{consequences}{Consequences}\dotfill \pageref*{consequences} \linebreak
\noindent\hyperlink{in_terms_of_indexed_categories}{In terms of indexed categories}\dotfill \pageref*{in_terms_of_indexed_categories} \linebreak
\noindent\hyperlink{in_the_internal_logic}{In the internal logic}\dotfill \pageref*{in_the_internal_logic} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \textbf{universe} in a [[topos]] is a [[topos theory|topos-theoretic]] version of the notion of [[Grothendieck universe]]; see that page for general motivation and applications.

To free the notion from membership-based set theory, we must replace \emph{sets of sets} by \emph{families of sets}, just as in passing from [[power set]]s to [[power object]]s we must replace sets of subsets by families of subsets.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

A \textbf{universe} in a topos $\mathcal{E}$ is a morphism $el\colon E \to U$ satisfying the axioms to follow. We think of $el\colon E \to U$ as an $U$-indexed family of objects/sets (fibers of $el$ being those objects), and we define a morphism $a\colon A \to I$ (regarded as an $I$-indexed family of objects) to be \textbf{$U$-small} if there exists a morphism $f\colon I \to U$ and a [[pullback]] square

\begin{displaymath}
\itexarray{A & \to & E\\
  ^a\downarrow && \downarrow^{el}\\
  I& \underset{f}{\to} & U.}
\end{displaymath}
The arrow $f$ is sometimes called the \textbf{name} of $a\colon A \to I$, since in the case when $I=*$, the arrow $f:* \to U$ \emph{points} at the term in the universe $U$ representing the object $A$. (See also this \href{http://nforum.ncatlab.org/discussion/6564/elementary-formulation-of-groupautomorphism-group/?Focus=53083#Comment_53083}{discussion} and references there.)

Note that $f$ is not, in general, unique: a universe can contain many isomorphic sets. With this definition, the pullback of a $U$-small morphism is automatically again $U$-small. We say that an object $X$ is $U$-small if $X\to 1$ is $U$-small.

The axioms which must be satisfied are:

\begin{enumerate}%
\item Every [[monomorphism]] is $U$-small.


\item The composite of $U$-small morphisms is $U$-small.


\item If $f\colon A \to I$ and $g\colon B \to A$ are $U$-small, then so is the [[dependent product]] $\Pi_f g$ (where $\Pi_f$ is the right adjoint to $f^*\colon \mathcal{E}/I \to \mathcal{E}/A$).


\item The [[subobject classifier]] $\Omega$ is $U$-small.



\end{enumerate}
Note that since $0\to \Omega$ is a monomorphism, (1) and (4) imply that the [[initial object]] $0$ is $U$-small. A \emph{[[predicative mathematics|predicative]] universe} is a morphism $el\colon E \to U$ where instead of (4) we assume merely that $0$ is $U$-small; this makes sense in any [[locally cartesian closed category]]. In a topos, the generic subobject $1\to \Omega$ is a predicative universe, and of course a morphism is $\Omega$-small if and only if it is a monomorphism.

If we assume only (1)--(3), then the identity morphism $1_0\colon 0 \to 0$ of the initial object would be a universe, for which it itself is the only $U$-small morphism. On the other hand, if $\mathcal{E}$ has a [[natural numbers object]] $N$, we may additionally assume that $N$ is $U$-small, to ensure that all universes contain ``infinite'' sets.

Note that any object isomorphic to a $U$-small object is $U$-small; thus in the language of [[Grothendieck universe]]s this notion of smallness corresponds to \emph{essential} smallness. Roughly, we may say that (1) corresponds to transitivity of a Grothendieck universe, (3) and (4) correspond to closure under power sets, and (2) corresponds to closure under indexed unions.

\hypertarget{example_universes_in_}{}\subsection*{{Example: universes in $SET$}}\label{example_universes_in_}

We spell out in detail some implications of these axioms for the case that the [[topos]] in question is the Categeory of Sets according to [[ETCS]], to be denoted $SET$.

Write $*$ for [[generalized the|the]] [[terminal object]] in $SET$, the singleton set. Notice that for each ordinary element $u \in U$, i.e. $* \stackrel{u}{\to} U$, there is the set $E_u$ over $u$, defined as the [[pullback]]

\begin{displaymath}
\itexarray{
  E_u &\to& E
  \\
  \downarrow && \downarrow
  \\
  * &\stackrel{u}{\to}& U
}
\end{displaymath}
We think of $E$ as being the disjoint union over $U$ of the $E_u$. In the language of indexed categories, this is precisely the case: the object $E\in SET$ is the indexed coproduct of the $U$-indexed family $(E\to U) \in SET/U$.

\begin{itemize}%
\item By the definition of $U$-smallness and the notation just introduced, an object $S$ in $SET$, regarded as a $*$-indexed family $S \to *$, is $U$-small precisely if it is isomorphic to one of the $E_u$.


\item If $S$ is a $U$-small set by the above and if $S_0 \hookrightarrow S$ is a [[monomorphism]] so that $S_0$ is a subset of $S$, it follows from 1) and 2) that the comoposite $(S_0 \hookrightarrow S \to *) = (S_0 \to *)$ is $U$-small, hence that $S_0$ is $U$-small. So: a subset of a $U$-small set is $U$-small.

\begin{itemize}%
\item In particular, let $\emptyset$ be the [[initial object]], which is a subset $\emptyset \hookrightarrow \Omega$ of $\Omega = \mathbf{2}$. So: the empty set is $U$-small.

\end{itemize}

\item Let $S$, $T$ and $K$ be objects of $SET$, regarded as $*$-indexed families $f\colon S \to *$, $T \to *$ and $K \to *$. Notice that $(SET\downarrow S)(f^* K, f^* T) \simeq (SET\downarrow S)(\itexarray{K \times S \\ \downarrow^{p_2} \\ S}, \itexarray{T \times S \\ \downarrow^{p_2} \\ S})$ is canonically isomorphic to $SET(K \times S, T)$. Since $\Pi_f$ is defined to be the [[right adjoint]] to $f^*\colon SET \to SET \downarrow S$ it follows that $\Pi_f f^* T \simeq T^S$ is the function set of functions from $S$ to $T$. By 3), if $S$, $T$ are $U$-small then so is the function set $T^S$.

\begin{itemize}%
\item Since by 4) $\Omega = \mathbf{2}$ is $U$-small and for every $S$ the function set $\mathbf{2}^S \simeq P(S)$ is the power set of $S$, it follows that the power set of a $U$-small set is $U$-small.

\end{itemize}

\item Let $I$ be a $U$-small set, in that $I \to *$ is $U$-small, and let $S \to I$ be $U$-small, to be thought of as an $I$-indexed family of $U$-small sets $S_i$, where $S_i$ is the [[pullback]] $\itexarray{
S_i &\to& S
\\
\downarrow && \downarrow
\\
* &\stackrel{i}{\to}& I
}$, so that $S$ is the disjoint union of the $S_i$: $S = sqcup_{i \in I} S_i$. By axiom 2) the composite morphism $(S \to I \to *) = (S \to *)$ is $U$-small, hence $S$ is a $U$-small set, hence the $I$-indexed union of $U$-small sets $\sqcup_{i \in I} S_i$ is $U$-small.



\end{itemize}
By standard constructions in [[set theory]] from these properties the following further closure properties of the universe $U$ follow.

\begin{itemize}%
\item For $I$ a $U$-small set and $S \to I$ an $I$-indexed family of $U$-small sets $S_i$, the cartesian product $\prod_{i \in I} S_i$ is $U$-small, as it is a subset of $P(I \times S)$.

\end{itemize}
In \textbf{summary}

\begin{itemize}%
\item the sets $\emptyset, *, \mathbf{2}$ are $U$-small;


\item a subset of a $U$-small set is $U$-small;


\item the power set $P(S)$ of any $U$-small set is $U$-small;


\item the function set $T^S$ for any two $U$-small sets is $U$-small;


\item the [[coproduct|union]] of a $U$-small family of $U$-small sets is $U$-small.


\item the [[product|product]] of a $U$-small family of $U$-small sets is $U$-small.



\end{itemize}
\hypertarget{axioms_of_universes}{}\subsection*{{Axioms of universes}}\label{axioms_of_universes}

Just as [[ZFC]] and other material [[set theory|set theories]] may be augmented with axioms guaranteeing the existence of [[Grothendieck universe]]s, so may [[ETCS]] and other structural set theories be augmented with axioms guaranteeing the existence of universes in the above sense. For example, the counterpart of Grothendieck's axiom

\begin{itemize}%
\item For every set $s$ there exists a universe $U$ containing $s$, i.e. $s\in U$

\end{itemize}
would be

\begin{itemize}%
\item For every morphism $a\colon A \to I$ in $\mathcal{E}$, there exists a universe $el\colon E \to U$ such that $a$ is $U$-small.

\end{itemize}
\hypertarget{consequences}{}\subsection*{{Consequences}}\label{consequences}

One can show, from the above axioms, that the $U$-small morphisms are closed under finite [[coproduct]]s and under [[quotient object]]s. See the reference below.

\hypertarget{in_terms_of_indexed_categories}{}\subsection*{{In terms of indexed categories}}\label{in_terms_of_indexed_categories}

Recall that an $\mathcal{E}$-[[indexed category]] is a [[pseudofunctor]] $\mathcal{E}^{op}\to \Cat$. The fundamental $\mathcal{E}$-indexed category is the \emph{self-indexing} $\mathbb{E}$ of $\mathcal{E}$, which takes $I\in \mathcal{E}$ to the [[over category|slice category]] $\mathbb{E}^I = \mathcal{E}/I$ and $x\colon I \to J$ to the [[base change]] functor $x^*$.

An \textbf{internal full subcategory} of $\mathcal{E}$ is a full sub-indexed category $\mathbb{F}$ of $\mathbb{E}$ (that is, a collection of full subcategories $\mathbb{F}^I\subset \mathbb{E}^I$ closed under reindexing) such that there exists a generic $\mathbb{F}$-morphism, i.e. a morphism $el\colon E \to U$ in $\mathbb{F}^U$ such that for any $a\colon A \to I$ in $\mathbb{F}^I$, we have $a \cong f^*(el)$ for some $f\colon I \to U$. In this case (since $\mathcal{E}$ is [[locally cartesian closed category|locally cartesian closed]]) there exists an [[internal category]] $U_1 \;\rightrightarrows\; U$ in $\mathcal{E}$ such that $\mathbb{F}$ is equivalent, as an indexed category, to the indexed category represented by $U_1 \;\rightrightarrows\; U$.

An internal full subcategory is an \textbf{internal full subtopos} if each $\mathbb{F}$ is a [[logical functor|logical]] subtopos of $\mathbb{E}$ (closed under finite limits, exponentials, and containing the subobject classifier). A universe in $\mathcal{E}$, as defined above, can then be identified with an internal full subtopos satisfying the additional axiom that $U$-small morphisms are closed under composition.

\hypertarget{in_the_internal_logic}{}\subsection*{{In the internal logic}}\label{in_the_internal_logic}

In a topos with a universe, we can talk about small objects in the [[internal logic]] by instead talking about elements of $U$. We can then rephrase the axioms of a universe in the internal logic to look more like the usual axioms for a Grothendieck universe, with the morphism $el\colon E \to U$ interpreted as a ``family of objects'' $(S_u)_{u\colon U}$:

\begin{enumerate}%
\item for all $u$ in $U$, if $X$ is a [[subset]] of $S_u$ (in the sense that there exists an [[injection]] $X \embedsin S_u$), then there is a $v$ in $U$ such that $X \cong S_v$;
\item for all $u$ in $U$, there is a $v$ in $U$ such that the [[power set]] $P(S_u) \cong S_v$;
\item there is a $v$ in $U$ such that $S_v$ is [[empty set|empty]];
\item for all $u$ in $U$ and all functions $f\colon S_u \to U$, there is a $v$ in $U$ such that the [[disjoint union]] $\biguplus_{i\colon S_u} S_{f(i)} \cong S_v$.

\end{enumerate}
In a [[well-pointed topos]], such as a model of [[ETCS]], these ``internal'' axioms are equivalent to their ``external'' versions that refer to [[global element]]s of $U$.

[[Mike Shulman|Mike]]: I haven't actually checked anything in this section, but it seems probable.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[type universe]]


\item [[object classifier in an (infinity,1)-topos]]


\item [[Grothendieck universe]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The construction of universes in presheaf toposes was first discussed in the following short note which might serve as a concise introduction to the more general reference that follows:

\begin{itemize}%
\item [[Martin Hofmann]], [[Thomas Streicher]], \emph{Lifting Grothendieck Universes} , ms. University of Darmstadt (unpublished). (\href{http://www.mathematik.tu-darmstadt.de/~streicher/NOTES/lift.pdf}{pdf})

\end{itemize}
The general case is considered here:

\begin{itemize}%
\item [[Thomas Streicher]], \emph{Universes in Toposes} , pp.78-90 in L. Crosilla, P. Schuster (eds.), \emph{From Sets and Types to Topology and Analysis: Towards Practicable Foundations for Constructive Mathematics} , Oxford UP 2005. (\href{http://www.mathematik.tu-darmstadt.de/~streicher/NOTES/UniTop.ps.gz}{ps},\href{http://www.mathematik.tu-darmstadt.de/~streicher/NOTES/UniTop.pdf}{pdf})

\end{itemize}
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